A225851 - OEIS

%I #11 May 17 2013 23:29:07

%S 3,3,7,7,7,19,59,59,59,59,157,13397,2312267,97760291,1042776437

%N For the Collatz (3x+1) iterations starting with a prime number p, a(n) is the smallest p such that the trajectory contains n successive prime numbers.

%C a(13)-a(15) were found by Farideh Firoozbakht. See Carlos Rivera's website. - _T. D. Noe_, May 17 2013

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_476.htm">Puzzle 476: A question related to Collatz sequence</a>

%e a(11) = 157 because the Collatz sequence of odd numbers is 157 -> 59 -> 89 -> 67 -> 101 -> 19 -> 29 -> 11 -> 17 -> 13 - > 5 -> 1 with 11 consecutive prime numbers.

%p nn:=300:T:=array(1..nn):

%p for n from 1 to 15 do:jj:=0:

%p for m from 2 to 10^5 while(jj=0) do:p:=ithprime(m):

%p for i from 1 to nn while(jj=0) do:

%p T[i]:=0:od:a:=1:T[1]:=p:x:=p:

%p for it from 1 to nn while (x>1) do:

%p if irem(x, 2)=0 then

%p x := x/2:

%p else

%p a:=a+1:T[a]:=x:

%p x := 3*x+1: fi:

%p od:

%p jj:=0:aa:=a:itr:=0:

%p for j from 2 to n+1 do:

%p if type(T[j], prime)=true then

%p itr :=itr+1 :

%p else fi:

%p od:

%p if itr=n then

%p jj:=1: printf ( "%d %d \n",n,p):

%p else

%p fi:

%p od:

%p od:

%t RemoveEven[n_] := n/2^IntegerExponent[n, 2]; Collatz2[n_] := NestWhileList[RemoveEven[3 # + 1] &, n, # > 1 &]; PrimeCnt[lst_] := Module[{i = 1}, While[PrimeQ[lst[[i]]], i++]; i - 1]; nn = 12; t = Table[0, {nn}]; found = 0; n = 2; While[found < nn, n = NextPrime[n]; ps = PrimeCnt[Collatz2[n]]; If[ps > nn, ps = nn]; While[ps > 0 && t[[ps]] == 0, t[[ps]] = n; found++; ps--]]; t (* _T. D. Noe_, May 17 2013 *)

%Y Cf. A006577.

%K nonn,hard

%O 1,1

%A _Michel Lagneau_, May 17 2013